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China TST 1999 any 3-element subset

Source: China TST 1999, problem 3

May 22, 2005
combinatorics unsolvedcombinatorics

Problem Statement

Let S={1,2,…,15}S = \lbrace 1, 2, \ldots, 15 \rbrace. Let A1,A2,…,AnA_1, A_2, \ldots, A_n be nn subsets of SS which satisfy the following conditions: I. ∣Ai∣=7,i=1,2,…,n|A_i| = 7, i = 1, 2, \ldots, n; II. ∣Ai∩Aj∣≤3,1≤i<j≤n|A_i \cap A_j| \leq 3, 1 \leq i < j \leq n III. For any 3-element subset MM of SS, there exists AkA_k such that M⊂AkM \subset A_k. Find the smallest possible value of nn.
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